{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,6]],"date-time":"2025-06-06T08:49:54Z","timestamp":1749199794838},"reference-count":13,"publisher":"American Mathematical Society (AMS)","issue":"351","license":[{"start":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T00:00:00Z","timestamp":1747094400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>For the Gauss\u2013Jacobi quadrature on <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-bracket negative 1 comma 1 right-bracket\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mo stretchy=\"false\">[<\/mml:mo>\n      <mml:mo>\u2212<\/mml:mo>\n      <mml:mn>1<\/mml:mn>\n      <mml:mo>,<\/mml:mo>\n      <mml:mn>1<\/mml:mn>\n      <mml:mo stretchy=\"false\">]<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">[-1,1]<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, the location is estimated where the kernel of the error functional for functions analytic on an ellipse and its interior in the complex plane attains its maximum modulus. For the Jacobi weight <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis 1 minus t right-parenthesis Superscript alpha Baseline left-parenthesis 1 plus t right-parenthesis Superscript beta\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mn>1<\/mml:mn>\n      <mml:mo>\u2212<\/mml:mo>\n      <mml:mi>t<\/mml:mi>\n      <mml:msup>\n        <mml:mo stretchy=\"false\">)<\/mml:mo>\n        <mml:mi>\u03b1<\/mml:mi>\n      <\/mml:msup>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mn>1<\/mml:mn>\n      <mml:mo>+<\/mml:mo>\n      <mml:mi>t<\/mml:mi>\n      <mml:msup>\n        <mml:mo stretchy=\"false\">)<\/mml:mo>\n        <mml:mi>\u03b2<\/mml:mi>\n      <\/mml:msup>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">(1-t)^\\alpha (1+t)^\\beta<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> (<inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha greater-than negative 1\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>\u03b1<\/mml:mi>\n      <mml:mo>&gt;<\/mml:mo>\n      <mml:mo>\u2212<\/mml:mo>\n      <mml:mn>1<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\alpha &gt;-1<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"beta greater-than negative 1\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>\u03b2<\/mml:mi>\n      <mml:mo>&gt;<\/mml:mo>\n      <mml:mo>\u2212<\/mml:mo>\n      <mml:mn>1<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\beta &gt;-1<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>) except for the Gegenbauer weight (<inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha equals beta\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>\u03b1<\/mml:mi>\n      <mml:mo>=<\/mml:mo>\n      <mml:mi>\u03b2<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\alpha =\\beta<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>), the location is the intersection point of the ellipse with the real axis in the complex plane. For the Gegenbauer weight, it is the intersection point(s) with either the real or the imaginary axis or other axes with angle <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"one fourth pi\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mstyle displaystyle=\"false\" scriptlevel=\"0\">\n        <mml:mfrac>\n          <mml:mn>1<\/mml:mn>\n          <mml:mn>4<\/mml:mn>\n        <\/mml:mfrac>\n      <\/mml:mstyle>\n      <mml:mi>\u03c0<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\tfrac {1}{4}\\pi<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> and <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"three fourths pi\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mstyle displaystyle=\"false\" scriptlevel=\"0\">\n        <mml:mfrac>\n          <mml:mn>3<\/mml:mn>\n          <mml:mn>4<\/mml:mn>\n        <\/mml:mfrac>\n      <\/mml:mstyle>\n      <mml:mi>\u03c0<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\tfrac {3}{4}\\pi<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. Our results support the empirical results provided by Gautschi and Varga [SIAM J. Numer. Anal, 20 (1983), pp.\u00a01170\u20131186] for the Jacobi weight, the Gegenbauer weight and the Legendre weight. The results obtained are also illustrated numerically.<\/p>","DOI":"10.1090\/mcom\/3977","type":"journal-article","created":{"date-parts":[[2024,5,11]],"date-time":"2024-05-11T01:58:06Z","timestamp":1715392686000},"page":"359-379","source":"Crossref","is-referenced-by-count":2,"title":["Error bounds for Gauss\u2013Jacobi quadrature of analytic functions on an ellipse"],"prefix":"10.1090","volume":"94","author":[{"given":"Hiroshi","family":"Sugiura","sequence":"first","affiliation":[]},{"given":"Takemitsu","family":"Hasegawa","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2024,5,13]]},"reference":[{"key":"1","unstructured":"B. C. Carlson, DLMF: Chapter 19, Elliptic Integrals, \\url{https:\/\/dlmf.nist.gov\/19\/}, 2024 (accessed January 15, 2024), NIST."},{"key":"2","series-title":"Computer Science and Applied Mathematics","isbn-type":"print","volume-title":"Methods of numerical integration","author":"Davis, Philip J.","year":"1984","ISBN":"http:\/\/id.crossref.org\/isbn\/0122063600","edition":"2"},{"key":"3","doi-asserted-by":"publisher","first-page":"352","DOI":"10.1553\/etna_vol53s352","article-title":"Error estimates of Gaussian-type quadrature formulae for analytic functions on ellipses\u2014a survey of recent results","volume":"53","author":"Djuki\u0107, D. Lj.","year":"2020","journal-title":"Electron. Trans. Numer. Anal."},{"key":"4","series-title":"Undergraduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-21607-2","volume-title":"Complex analysis","author":"Gamelin, Theodore W.","year":"2001","ISBN":"http:\/\/id.crossref.org\/isbn\/0387950931"},{"issue":"6","key":"5","doi-asserted-by":"publisher","first-page":"1170","DOI":"10.1137\/0720087","article-title":"Error bounds for Gaussian quadrature of analytic functions","volume":"20","author":"Gautschi, Walter","year":"1983","journal-title":"SIAM J. Numer. Anal.","ISSN":"http:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"6","unstructured":"T. H. Koornwinder, R. Wong, R. Koekoek, and R. F. 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